Of course, one does not need to rely on pattern analysis to discover this. It follows directly from the paper's definition of the number for a given graph. The graphs in which the root has order 1 are exactly those whose associated numbers have exactly one prime factor (of order 1), and that factor is the nth prime, where n is the number assigned to the subgraph obtained by deleting the root of the original and choosing its adjacent vertex as the root of the new graph.